Hilbert's irreducibility theorem
# Hilbert's irreducibility theorem

In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible. This theorem is a prominent theorem in number theory.

## Formulation of the theorem

Hilbert's irreducibility theorem. Let

    $f_{1}(X_{1},\ldots ,X_{r},Y_{1},\ldots ,Y_{s}),\ldots ,f_{n}(X_{1},\ldots ,X_{r},Y_{1},\ldots ,Y_{s})$

be irreducible polynomials in the ring

    $\mathbb {Q} (X_{1},\ldots ,X_{r})[Y_{1},\ldots ,Y_{s}].$

Then there exists an r-tuple of rational numbers $(a_1,$ ..., $a_r)$ such that

    $f_{1}(a_{1},\ldots ,a_{r},Y_{1},\ldots ,Y_{s}),\ldots ,f_{n}(a_{1},\ldots ,a_{r},Y_{1},\ldots ,Y_{s})$

are irreducible in the ring

    $\mathbb {Q} [Y_{1},\ldots ,Y_{s}].$

Remarks.

* It follows from the theorem that there are infinitely many r-tuples. In fact the set of all irreducible specializations, called Hilbert set, is large in many senses. For example, this set is Zariski dense in $\mathbb {Q} ^{r}.$
* There are always (infinitely many) integer specializations, i.e., the assertion of the theorem holds even if we demand $(a_1,$ ..., $a_r)$ to be integers.
* There are many Hilbertian fields, i.e., fields satisfying Hilbert's irreducibility theorem. For example, number fields are Hilbertian.
* The irreducible specialization property stated in the theorem is the most general. There are many reductions, e.g., it suffices to take $n=r=s=1$ in the definition. A result of Bary-Soroker shows that for a field K to be Hilbertian it suffices to consider the case of $n=r=s=1$ and $f=f_{1}$ absolutely irreducible, that is, irreducible in the ring $K^{alg}[X,Y],$ where $K^{alg}$ is the algebraic closure of K.

## Applications

Hilbert's irreducibility theorem has numerous applications in number theory and algebra. For example:

* The inverse Galois problem, Hilbert's original motivation. The theorem almost immediately implies that if a finite group G can be realized as the Galois group of a Galois extension N of

       $E=\mathbb {Q} (X_{1},\ldots ,X_{r}),$
    then it can be specialized to a Galois extension $N_0$ of the rational numbers with G as its Galois group. (To see this, choose a monic irreducible polynomial $f(X_1,$ ..., $X_n,$ Y) whose root generates N over E. If $f(a_1,$ ..., $a_n,$ Y) is irreducible for some $a_i,$ then a root of it will generate the asserted $N_0.)$
* Construction of elliptic curves with large rank.
* Hilbert's irreducibility theorem is used as a step in the Andrew Wiles proof of Fermat's Last Theorem.
* If a polynomial $g(x)\in \mathbb {Z} [x]$ is a perfect square for all large integer values of x, then g(x) is the square of a polynomial in $\mathbb {Z} [x].$ This follows from Hilbert's irreducibility theorem with $n=r=s=1$ and

       $f_{1}(X,Y)=Y^{2}-g(X).$
    (More elementary proofs exist.) The same result is true when "square" is replaced by "cube", "fourth power", etc.

## Generalizations

It has been reformulated and generalized extensively, by using the language of algebraic geometry. See thin set (Serre).

